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Summary

This book is for sophomore-level or junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite.

This thorough and accessible text from one of the leading figures in the use of technology in linear algebra gives students a challenging and broad understanding of the subject. The author infuses key concepts with their modern practical applications to offer students examples of how mathematics is used in the real world.

Each chapter contains integrated worked examples and chapter tests. The book stresses the important roles geometry and visualization play in understanding linear algebra. This edition will continue to be packaged with the ancillary ATLAST computer exercise guide, as well as new MATLAB and Maple guides, which also come with the package.

One of the longer sections in the previous edition was the section on matrix algebra in Chapter 1. The material in that section has been expanded further for the current edition. Rather than include an overly long revised section, we have divided the material into sections titled Matrix Arithmetic and Matrix Algebra.

2. New Exercises

After seven editions it was quite a challenge to come up with additional original exercises. This eighth edition has more than 130 new exercises. The new exercises are not evenly distributed throughout the book. Some sections have many new exercises and others have few or none.

3. New Subsections and Applications

A new subsection on cross products has been included in Section 3 of Chapter 2. A new application to Newtonian Mechanics has also been added to that section. In Section 4 of Chapter 6 (Hermitian Matrices), a new subsection on the Real Schur Decomposition has been added.

4. New and Improved Notation

The standard notation for the jth column vector of a matrix A is aj , however, there seems to be no universally accepted notation for row vectors. In the MATLAB package, the ith row of A is denoted by A(i, :). In previous editions of this book we used a similar notation a(i, :), however, this notation seems somewhat artificial. For this edition we use the same notations as for a column vector except we put a horizontal arrow above the letter to indicate that the vector is a row vector (an horizontal array) rather than a column vector (a vertical array).

We have also introduced improved notation for the standard Euclidean vector spaces and their complex counterparts.

5. Other Revisions

Various other revisions have been made throughout the text. Many of these revisions were suggested by reviewers.

6. Special Web Site and Supplemental Web Materials

Pearson has developed a special Web site to accompany the 8th edition. This site includes a host of materials for both students and instructors.

Author Biography

Steven J. Leon is a Chancellor Professor of Mathematics at the University of Massachusetts Dartmouth. He has been a Visiting Professor at Stanford University, ETH Zurich (the Swiss Federal Institute of Technology), KTH (the Royal Institute of Technology in Stockholm), UC San Diego, and Brown University. His areas of specialty are linear algebra and numerical analysis.

Leon is currently serving as Chair of the Education Committee of the International Linear Algebra Society and as Contributing Editor to Image, the Bulletin of the International Linear Algebra Society. He had previously served as Editor-in-Chief of Image from 1989 to 1997. In the 1990’s he also served as Director of the NSF sponsored ATLAST Project (Augmenting the Teaching of Linear Algebra using Software Tools). The project conducted 18 regional faculty workshops during the period from 1992–1997.

Table of Contents

Preface

What's New in the Eighth Edition?

Computer Exercises

Overview of Text

Suggested Course Outlines

Supplementary Materials

Acknowledgments

Matrices and Systems of Equations

Systems of Linear Equations

Row Echelon Form

Matrix Arithmetic

Matrix Algebra

Elementary Matrices

Partitioned Matrices

Matlab Exercises

Chapter Test A

Chapter Test B

Determinants

The Determinant of a Matrix

Properties of Determinants

Additional Topics and Applications

Matlab Exercises

Chapter Test A

Chapter Test B

Vector Spaces

Definition and Examples

Subspaces

Linear Independence

Basis and Dimension

Change of Basis

Row Space and Column Space

Matlab Exercises

Chapter Test A

Chapter Test B

Linear Transformations

Definition and Examples

Matrix Representations of Linear Transformations

Similarity

Matlab Exercises

Chapter Test A

Chapter Test B

Orthogonality

The Scalar Product in Rn

Orthogonal Subspaces

Least Squares Problems

Inner Product Spaces

Orthonormal Sets

The Gram-Schmidt Orthogonalization Process

Orthogonal Polynomials

Matlab Exercises

Chapter Test A

Chapter Test B

Eigenvalues

Eigenvalues and Eigenvectors

Systems of Linear Differential Equations

Diagonalization

Hermitian Matrices

The Singular Value Decomposition

Quadratic Forms

Positive Definite Matrices

Nonnegative Matrices

Matlab Exercises

Chapter Test A

Chapter Test B

Numerical Linear Algebra

Floating-Point Numbers

Gaussian Elimination

Pivoting Strategies

Matrix Norms and Condition Numbers

Orthogonal Transformations

The Eigenvalue Problem

Least Squares Problems

Matlab Exercises

Chapter Test A

Chapter Test B

Appendix: MATLAB

The MATLAB Desktop Display

Basic Data Elements

Submatrices

Generating Matrices

Matrix Arithmetic

MATLAB Functions

Programming Features

M-files

Relational and Logical Operators

Columnwise Array Operators

Graphics

Symbolic Toolbox

Help Facility

Conclusions

Bibliography

Linear Algebra and Matrix Theory

Applied and Numerical Linear Algebra

Books of Related Interest

Answers to Selected Exercises

Table of Contents provided by Publisher. All Rights Reserved.

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Customer Reviews

An easy understanding of Linear AlgebraAugust 8, 2011

by William Byrd

A very good undergraduate textbook dealing with Linear Algebra. The text is very clear on the subject. But, the real strength of this textbook is the examples is using the MatLab computer program. I have been able to duplicate the problems using Octave computer program.

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Linear Algebra With Applications: 4 out of 5 stars based on 1 user reviews.